Headnote
ABSTRACT
Objective: The objective of this study is to investigate a machine learning methodology based on the physics of dynamic fluid flow, with the aim of incorporating the knowledge of physical laws into learning algorithms to estimate precise results with lower computational cost.
Theoretical Framework: This section presents the main concepts and theories that underpin the research. Key theories include nonlinear partial differential equations (PDEs), Darcy's Law, and the conservation of mass and energy, providing a solid foundation for understanding the research context.
Method: The methodology adopted for this research involves modeling a one-dimensional two-phase fluid flow problem in a porous medium, governed by a first-order hyperbolic nonlinear equation. Data collection was conducted through numerical simulations, using micro and macro grids to evaluate permeability and boundary conditions.
Results and Discussion: The results revealed that considering learning only in absolute permeability yielded good estimates of reservoir pressures. However, small discrepancies were observed in the estimation of saturations. The discussion contextualizes these results in light of the theoretical framework, highlighting the identified implications and relationships, as well as the study's limitations.
Research Implications: The practical and theoretical implications of this research are discussed, providing insights into how the results can be applied or influence practices in the field of reservoir modeling and fluid flow simulations. These implications may include optimizing oil extraction processes and improving predictive models in reservoir engineering.
Originality/Value: This study contributes to literature by presenting an innovative approach that integrates physical laws into machine learning algorithms, resulting in more accurate and efficient estimates. The relevance and value of this research are evidenced by the potential application of the results in optimizing industrial processes and improving computational simulations.
Keywords: Machine Learning, Physics-Based Modeling, Fluid Flow, Porous Medium, Reservoir.
RESUMO
Objetivo: O objetivo deste estudo é investigar uma metodologia de aprendizagem de máquina baseada na física do fluxo dinámico de fluidos, com o objetivo de incorporar o conhecimento de leis físicas em algoritmos de aprendizagem para estimar resultados precisos com menor custo computacional.
Estrutura Teórica: Esta seção apresenta os principais conceitos e teorias que sustentam a pesquisa. As principais teorias incluem equações diferenciais parciais não lineares (EDPs), a Lei de Darcy e a conservação de massa e energia, fornecendo uma base sólida para a compreensão do contexto de pesquisa.
Método: A metodologia adotada para esta pesquisa envolve a modelagem de um problema de fluxo de fluido bifásico unidimensional em um meio poroso, governado por uma equacáo hiperbólica náo-linear de primeira ordem. A coleta de dados foi realizada por meio de simulações numéricas, utilizando microrredes e macroredes para avaliar a permeabilidade e as condições de contorno.
Resultados e Discussão: Os resultados revelaram que considerar o aprendizado apenas em permeabilidade absoluta produziu boas estimativas de pressões de reservatório. No entanto, foram observadas pequenas discrepâncias na estimativa de saturaçôes. A discussão contextualiza esses resultados a luz do marco teórico, destacando as implicações e relações identificadas, bem como as limitações do estudo.
Implicações da pesquisa: As implicações práticas e teóricas desta pesquisa são discutidas, fornecendo insights sobre como os resultados podem ser aplicados ou influenciar práticas no campo da modelagem de reservatórios e simulações de fluxo de fluidos. Essas implicações podem incluir a otimização dos processos de extração de petróleo e a melhoria dos modelos preditivos na engenharia de reservatórios.
Originalidade/valor: Este estudo contribui para a literatura, apresentando uma abordagem inovadora que integra as leis físicas em algoritmos de aprendizagem de máquina, resultando em estimativas mais precisas e eficientes. A relevância e o valor desta pesquisa são evidenciados pela aplicação potencial dos resultados na otimização de processos industriais e na melhoria de simulações computacionais.
Keywords: Aprendizagem de Máquina, Modelagem Baseada em Física, Fluxo de Fluidos, Meio Poroso, Reservatorio.
RESUMEN
Objetivo: El objetivo de este estudio es investigar una metodología de aprendizaje automático basada en la física del flujo dinámico de fluidos, con el objetivo de incorporar el conocimiento de las leyes físicas en algoritmos de aprendizaje para estimar resultados precisos con menor costo computacional.
Marco teórico: Esta sección presenta los principales conceptos y teorías que sustentan la investigación. Las teorías clave incluyen ecuaciones diferenciales parciales (PDEs) no lineales, la Ley de Darcy y la conservación de masa y energía, proporcionando una base sólida para entender el contexto de investigación.
Método: La metodología adoptada para esta investigación consiste en modelar un problema unidimensional de flujo de fluido bifásico en un medio poroso, gobernado por una ecuación hiperbólica no lineal de primer orden. La recolección de datos se llevó a cabo mediante simulaciones numéricas, utilizando microrredes y macroredes para evaluar la permeabilidad y las condiciones de frontera.
Resultados y Discusión: Los resultados revelaron que considerar el aprendizaje solo en permeabilidad absoluta produjo buenas estimaciones de las presiones del reservorio. Sin embargo, se observaron pequeñas discrepancias en la estimación de las saturaciones. La discusión contextualiza estos resultados a la luz del marco teórico, destacando las implicaciones y relaciones identificadas, así como las limitaciones del estudio.
Implicaciones de la investigación: Se discuten las implicaciones prácticas y teóricas de esta investigación, aportando ideas sobre cómo los resultados pueden ser aplicados o influir en prácticas en el campo de la modelación de yacimientos y simulaciones de flujo de fluidos. Estas implicaciones pueden incluir la optimización de los procesos de extracción de petróleo y la mejora de los modelos predictivos en la ingeniería de los yacimientos.
Originalidad/Valor: Este estudio contribuye a la literatura presentando un enfoque innovador que integra leyes físicas en algoritmos de aprendizaje automático, lo que resulta en estimaciones más precisas y eficientes. La relevancia y el valor de esta investigación se evidencian en la aplicación potencial de los resultados en la optimización de procesos industriales y en la mejora de simulaciones computacionales.
Palabras clave: Aprendizaje Automático, Modelado Basado en la Física, Flujo de Fluidos, Medio Poroso, Depósito.
1 INTRODUCTION
Recently, techniques derived from Machine Learning have gained great visibility in the areas of computer science and engineering communities, achieving remarkable results in their applications in supervised, semi-supervised and unsupervised learning. However, for the effective use of these techniques, a large amount of data with the necessary quality 1s required for the development and solution of the applications under analysis, which makes the various engineering applications difficult due to the costs associated with the acquisition of this data.
Problems related to computational geomechanics in underwater dynamic flow modeling suffer mainly from data quality. One way that, recently, has gained a lot of visibility in helping this problem, is the contribution of additional information about the Laws of Physics, such as Conservation of Mass, Conservation of Energy, Darcy's Law, etc. In fact, the incorporation of this information in computational mechanics problems allows learning algorithms to solve partial differential equations (PDE's), thus proposing an alternative solution to traditional methods of solving differential equations, such as finite differences and finite volumes.
The main objective of incorporating the Laws of Physics into learning algorithms is to use them as a priori information so that relevant information can be extracted for predicting a solution with adequate precision and accuracy.
2 MULTIPHASE FLOW PROBLEM
Consider the standard Buckley-Leverett model with two incompressible and immiscible fluids, for example: oil and water. Being the oil (0), "non-wet" phase, displaced by water (w), wet phase, in a porous medium with permeability k(x) and porosity (x). Gravity and capillarity neglected.
Under these conditions, the pressure (P) and saturation of fluids Sy (а = 0,W) are governed by a system of mass balance equations complemented by Darcy's Law for each of the phases. Introducing the total flow system (Uzoz) in incompressible form:
... (1)
Where q; is the source/sink term, and the conservation equation for one of the phases is:
- Water
... (2)
Where:
Utot = Uy + uo 15 the total flux and ug represents the Darcy flux for each phase (а = o, w); the function fw is the fractional flow of water, and is defined as:
- Water
... (3)
Where:
*q = (k · kra )/·a is the phase mobility, ug is the phase viscosity, k,.q(S,) is the phase relative permeability and q, is the phase source or sink term, which represents the effects of the wells. Equations (2) require boundary conditions, given by:
... (4)
... (5)
Where:
Sy 15 the initial water saturation in the reservoir, and 5, is the saturation in the well injection or in the contour Гу.
In the one-dimensional case, Equation (2) becomes:
... (6)
Since the total speed иго; is a constant. After introducing the dimensionless variables tp = J (or dt")/®L] and xp = x/L, where Lis the length of the one-dimensional system, then, rewriting Equation (6):
... (7)
And the initial and boundary conditions, then:
... (8)
... (9)
Solving the presented initial value problem is the same as solving the hyperbolic
nonlinear equation of Equation (9), below:
... (10)
With the initial condition constant:
... (11) Where:
u(t, x), is the quantity of interest that depends on the spacetime that needs to be solved and f (u) is the flux function.
3 PHYSICS BASED LEARNING
Initially, based on the Distortion Rate Theory proposed by Shannon and Kolmogorov (Cover, 2006), we use the distortion rate optimization problem given in (12) with the restriction defined by the mathematical expectation of the distortion. Where X is the original random field (micromesh) quantized by the sequence of realizations (x;); ·, each with its probability p(x), and Y the compressed random field (macromesh) and quantized by its realizations (y; HR with associated probabilities p(y) = Yr e x P(Y|X)p(x) , então:
... (12)
Being, ... = ... = ... Graphically illustrating the Distortion Rate Function:
Finding the distortion rate function is a variational problem, which can be solved with the aid of a Lagrange multiplier, В. So, to impose the constraint on distortion, it remains to minimize the functional:
... (13)
over all distributions p(y|x), consistent, that is Y, cy p(y|x) = 1. This last condition requires one more Lagrange multiplier A(x) for each x; the functional be to extreme becomes:
... (14)
where both ß and 4 must not be negative. The problem declared by functional (13) can also be solved by its complementary functional:
... (15)
The functionals F and F" are simultaneously extreme for the same density p(y|x) and for · =T 1.
The solution to the problem is given by the following well-known result:
Theorem 1 (Arimoto, 1972; Cover, 1991, 2006; Tishby, Pereira and Bialek, 1999). The solution to the variational problem
... (16)
for consistent p(y, x) densities, it is given by the exponential form
... (17)
where Z(x, ·) is a normalization function given by
... (18)
Furthermore, the Lagrange multiplier 3, determined by the mathematical expectation value of the distortion, D, is positive and satisfies
... (19)
where:
R is given by equation (12).
3.1 ALGORITMO'S ALGORITHM - SUCCESSIVE SUBSTITUTIONS
A consequence of the variational formulation discussed and summarized in Theorem 1 is that it provides an iterative algorithm, Arimoto (1972) and Blahut (1972), for the consistent determination of densities p(y|x) and p(y), given by equations p(y) and (17). Thus, the algorithm determines the optimal partition of support for the realizations, but not the representative values of Y.
However, it is also essential to determine those representative values, for a given value of 6, as well as the optimal value that minimizes the distortion, given the partition. The Deterministic Annealing (DA) algorithm, used here and discussed below, solves both problems.
The problem consists of satisfying equations p(y) and (17) in a self-consistent way. A natural process is to solve those equations by successive substitutions, up to a pre-established precision. The following auxiliary lemma establishes the global convergence for this case. Global convergence 1s a highly desirable property of some algorithms to approximate the desired answer as closely as desired for any and all starting points.
Lemma 2. (Csiszár and Tusnady, 1984). Let p(x, y) = p(y|x)p(x) be a joint distribution.
Then, the p(y) distribution that minimizes the Relative Entropy or KL divergence, Dy; is the marginal
... (20)
Specifically,
... (21)
Equivalently, the distribution q(y) that minimizes the mathematical expectation of entropy,
... (22)
Is also the marginal p(y) = Y xx PY |x) p(x).
The proof of this lemma is a direct consequence that relative entropy is not negative. He guarantees that equations p(y) and (17) come from the same variational principle.
Theorem 3. (Arimoto's Algorithm, 1972). Equations p(y) and (17) are satisfied simultaneously in the minimum of the functional, equation (14),
... (23)
where minimization is done independently over convex sets of consistent densities, {p(y)}e{p(y|x)},
min min ... (24)
These independent conditions correspond precisely to equations p(y) and (17). Denoting an iterative step by k,
... (25)
where the normalization function Zi (x, В) is updated at each К iteration by equation (18). Furthermore, those iterations converge to a single minimum of the functional F, on the convex set of the two densities.
For proof of this theorem, see Cover (2006). As noted earlier, quantized values, in particular optimal values that minimize distortion, will be obtained with the aid of DA.
3.2 ROSE ALGORITHM - DETERMINISTIC ANNEALING
Deterministic Annealing (DA, Deterministic Annealing) is the deterministic version of Simulated Annealing (SA, Simulated Annealing) which takes its name due to its analogy to the physical-chemical annealing process of materials, in which the controlled cooling process begins at high temperatures in such a way as to increase the size of the crystals and reduce their defects, while maintaining thermal equilibrium. The optimal joint solution, being the Gibbs distribution, makes SA an attractive algorithm.
SA is used to find the global minimum of a function with a large number of variables, as in Equilibrium Statistical Mechanics, in the quantization process, when precision is not a fundamental item. Within the cooling cycles there is a stochastic algorithm (usually the Metropolis's algorithm or some variant) for determining the equilibrium configuration at each temperature. With each new temperature, the solution, or equilibrium state, eventually increases the number of quanta (realizations). This, therefore, would be an ideal method for the solution process proposed in this work, were it not for its imprecision in the values and associated probabilities of the quantization. This imprecision is an inherent quality of stochastic algorithms, which can be solved by hybridization with a deterministic algorithm (Fonseca, 2015).
DA (Rose, 1991, 1994, 1998) does this hybridization by replacing the stochastic algorithm inside the SA with the steepest descent (SD), producing a new algorithm with much better convergence properties.
Contrary to SA, there is no proof of convergence to the global minimum, in the general case, but only of global convergence. However, as previously established, each DA local problem (each temperature or B value) used in this work is convex, and the extremization of the functional is also a convex problem, according to Theorem 3.
The crucial aspect in DA is the determination of the optimal functional gradient with respect to each sample ук. For this, it is convenient to use the complementary functional, given by equation (15),
... (26)
Substituting the optimal density, of Gibbs, in this functional, we obtain,
... (27)
It is convenient to write the optimal distribution, equation (18), as,
... (28)
The gradient to be used in the steepest descent method is obtained from the following minimum necessary condition, for a given macrorealization (terms with dx disappear due to the Kuhn-Tucker condition),
... (29)
For the case of distances given by the equation d(x, у) = (R (y) - r(x)', equation (29), takes the form,
... (30)
resulting in,
... (31)
where,
... (32)
and, finally,
... (33)
The inverse problem in the macroelement declared in equation (33) has only one dimension, one variable, however, in some situations of interest, it is not a linear problem, requiring an iterative process, with one variable, in its solution.
It should be noted, in equation (33), that the realization value depends directly on the joint and conditioned mathematical expectation, Ej (x,y,), of the microscale response, inversely weighted by the quantum probability, p(y).
The right alternative to the analytical solution of equation (33) is to use the steepest . . . : : . . д descent in numerical form using equation (30), precisely calculating the gradient, GE Note, k however, that, since the problem has only one variable, it should suffice to estimate the gradient by finite differences with high precision.
4 APPLICATIONS
1- In this example, the proposed methodology is applied to a one-dimensional reservoir with two-phase flow, oil and water, and depletion by water injection, studied by Emerick and Reynolds (2013).
The reservoir has 31 blocks of dimensions 50ft x 50ft x 50ft. The natural logarithm of the permeabilities, In(k), has an exponential autocorrelation function with a correlation length of 10 blocks, with mean 5 and variance 1. Porosity is constant and equal to 0.25, water viscosity is 1cP and that of oil, 2cP. The initial pressure in the reservoir is 3500psi and the compressibility of oil, water and rock are 10 psi 1,10 psi + e 5x10 "psi! respectively. There is a water injection well in the first block that operates with a bottom well pressure of 4000psi. In the last block there is a producing well that operates at a bottomhole pressure of 3000psi. There is a pressure observation well in the center of the reservoir. The production period is 360 days, with monthly measurements. The production period was defined such that there is water production in the observation well, but not in the producing well. The micromesh is shown in figure 2.
The authors generated a permeability field described by 1000 equally probable realizations, based on the scenario shown in figure 2. All the problem data were provided by the authors through digital files, as well as the executable code of the simulator used by them. The problem was originally devised to compare methods for solving inverse problems for permeability fields.
The exercise here consists of determining the new absolute permeability field for the macro-grid given in Figure 3. In it, the original dimensions of the three blocks with wells were maintained. The other blocks were scaled two by two, that is, with a = 2, resulting in a macromesh of 17 blocks.
The reservoir simulations used in this example were carried out with the MRST-2017b, Lie (2016), and the scale transposition was carried out with the values taken from the simulations, in a single-phase problem (water), when the velocities were in a regime stationary.
The procedure starts with the simulation of 1000 realizations in the microgrid. These are the only simulations performed on the microgrid. Fourteen scale changes were performed, one for each macroelement, with Dirichlet boundary conditions in the simulation of each macroelement, and with the distortion measure reported in Appendix A.
The boundary conditions for each macroelement were calculated with the stationary values of the microgrid, where the left (right) pressure 1s the pressure of the microelement to its left (right).
The answers, r(x), reported in Appendix A were the interface velociites od each two microelements; while the answers, В (ук), are the velocities at the center of each macroelement. For the solution of the one variable inverse problem, given by Appendix A,
... 34
Where:
r(x) is the average velocity at the center of the macroelement, и is the viscosity of the fluid, and VP is the pressure gradient of the macroelement of the macroelement, respectively.
Table 1, in Appendix C, summarizes the data and results for this example. Figure 4 demonstrates the Medium distortion and the Mutual information obtained by the algorithm, the quantization of the p.d.f.'s of the absolute permeability for two macroelements and shows the velocities obtained for each element of the macrogrid and microgrid.
With the objective of verifying the results obtained from the methodology with the consideration of the laws of physics, a comparison between the model obtained by the macrogrid and the original model in the microgrid, both biphasic, was carried out. Figures 6 and 7 shows water and oil saturations and reservoir pressures at 150, 300, 450 and 750 days in both the macro and micro meshes.
From the results obtained in Table 1, it can be seen that the Din were very small, with an order of magnitude of 107°, a fact that results from the application of the PDA algorithm in just two blocks. Mutual Information, a learning indicator, showed good results for non-linear problems.
The number of realizations, np, varied between 26 and 35, which demonstrates a very significant reduction in the probabilistic amount of the micromesh for the input parameters used, in the order of 95%.
From the analysis of the graphs of the saturations and pressures of the biphasic reservoir, it is possible to affirm that the results obtained with 1000 realizations of the micromesh and 10 realizations of the macromesh are very similar, demonstrating that, with the application of the procedures developed in Appendix A in problems of biphasic reservoirs, it is It is possible to reduce both the probabilistic dimension and the cardinality of the meshes in a controlled and goal-oriented way.
2- In this example, the same data as in example 1 were used, with the aim of obtaining a macro-mesh that represents the reservoir under study. However, the procedures developed in Appendix B were adopted, in which the upscaling is performed for the relative permeabilities of the oil and water phases. The procedure adopted was carried out at each time step of the reservoir simulation.
As in example 1, the responses, r(x), reported in Appendix B were the velocities at the interface of each two microelements, being i the oil phase (о) or the water phase (w); while the responses, R(Yx;), are the velocities at the center of each macroelement, being, i the oil phase (0) or the water phase (w).
For the solution of the one-variable inverse problem, given by Appendix B,
... (35)
Where:
r(x;) is the average velocity at the center of the macroelement, и; is the viscosity of the fluid and VP is the pressure gradient of the macroelement, respectively, where i is the oil phase (0) or the water phase (w). Therefore, from Appendix B
For oil phase:
... (36)
For water phase:
... (37)
Tables 2 and 3, in Appendix C, summarize some the data and results for this example. Figure 7 and 9 shows the Average Distortion and Mutual Information, demonstrates the quantization of the p.d.f.'s of the relative permeability for the water and oil phases obtained during the application of the methodology presented in Appendix B for two macroelements and shows the velocities obtained of the macromesh and micromesh.
From table 4 and 5, in Appendix C, we see a greater number of realizations, ng, for the final macroelements, which can be attributed to the very characterization of the reservoir, with the location of the oil producing well close to the last macroelement and the variability of the relative permeability of the oil phase.
A minimum mean distortion is observed, with an order of magnitude from 1078 a 10712, which characterizes an excellent reduction of errors for the estimates of the realization values of the relative permeability of the oil phase in the macromesh.
The Mutual Information learning indicator was in a range of 0.02 to 0.89, which can be attributed to the local variability of permeability in the microgrid. The value 0, indicated in the initial macroelements, is attributed to the little variability of the Darcy velocity in the oil phase throughout the reservoir simulation, since the water injection well is located close to it.
In contrast to the oil phase, in table 3, in Appendix C, we see a greater number of realizations, ng, for the initial macroelements of the reservoir, which is attributed to the very characterization of the reservoir, with the location of the water injection well close to the first macroelement and the variability of the relative permeability of the water phase.
As in the oil phase, a minimum average distortion is observed, with an order of magnitude from 1078 a 10711, which reinforces the excellent reduction of errors for the estimates of values of the realizations of the relative permeability of the water phase in the macro mesh.
The Mutual Information learning indicator was also in a range of 0.015 to 1.60, with the exception of some lower peaks, which can be attributed to the local variability of permeability in the microgrid.
The simulation during the entire useful life of the reservoir contributed to the analysis of the application of the algorithm in the two-phase reservoir under study. In both phases, it is verified that the algorithm was able to capture the variability necessary for the representativeness of the relative permeability realizations in all macroelements. However, in the water phase, the algorithm captured more realizations in the initial macroelements, which is consistent, due to the location of the injection well. In the oil phase, as the producing well is located at the final end of the macroelements, the highest capture of realizations was close to the last macroelement. This fact is attributed to the variability of the data found in the simulation of the reservoir in regions of wells.
5 CONCLUSION AND FUTURE WORK
A physics-based machine learning methodology was presented in a computational geomechanics problem of submerged two-phase dynamic flow modeling, in which very satisfactory and encouraging results were obtained. In example 1, the upscaling of the absolute permeabilities was performed, as described in Appendix A. Next, the simulation of the reservoir was performed with the data determined by the learning algorithm. The comparison, at different simulation times, was very representative. As for example 2, we present an embryonic proposal, in appendix B, for carrying out the upscaling of relative permeabilities. However, as in example 1, the comparative results between the Darcy velocities in the macrogrid and in the microgrid were very representative, which allows us to go further.
In future developments, it is recommended to insert other distortion measures and objectives for the comparison of the reservoir in all its exploitation.
ACKNOWLEDGEMENTS
The Authors acknowledge the Agéncia Nacional de Petróleo, Gás Natural e Biocombustíveis (ANP) and Financiadora de Estudos e Projetos (FINEP), by the financial support provided by the Programa de Recursos Humanos da ANP - PRH-47.1 supported by investments by Petroleum firms qualified in Cláusula de P, D&I of Resolução ANP n 50/2015, with support to scholarships and complementary funds for realizations of research and professional formation in Geology and Civil Engineering for the Petroleum and Gas Industries.
APPENDIX A. FORMULATIONS DEVELOPED FOR APPLICATION 1
In the case presented in application 1, we developed the methodology described in section 3, with the objective of performing the compression, upscaling, of the absolute permeabilities and obtaining a macromesh with lower information density, but with the necessary precision and accuracy for the adequate representation of the problem. Thus, the upscaling of the absolute permeability develops as:
1. у = f(Kaps);
2. Average Distortion: d(x, у) = (r (y) - ro)", where r(·) is the Darcy's velocities.
3. Responses: r(y) = - - where и is the fluid viscosity, VP the pressure gradient and y the absolute permeabilities in macromesh.
After the initial definitions, from Eq. (20), we were able to develop an equation that represents the absolute permeabilities for the problem under analysis:
...,
...,
...,
...,
...
Observing, that:
1. У хех P(x) P(Yr|x) with respect to y, returns the probabilities associated with y, i.e., 40);
2. Y xex P(x) р(ук|х) with respect x, returns the medium value, ie;
So:
...
...
...
As the algorithm treats the probability of information regardless of the realizations, then in estimating them, we have:
...
APPENDIX B. FORMULATIONS DEVELOPED FOR APPLICATION 2
In application 2, a methodology similar to application 1 was developed, however the objective was to perform the compression, upscaling, of the relative permeabilities for the oil and water phases, still without considering the variable time in the formulations, due to the introductory nature of the formulations. Therefore, we start the upscaling by defining the
equations used:
1. у = f(K relagua' ; Kretotco);
2. Average Distortion: d(x,y) = (r (y) - r(x)', where r(m) is the Darcy's velocities.
3. Responses: r (y) = - abs Y E where и is the fluid viscosity, VP the pressure gradient,
Kips the absolute permeability in macromesh and y the relative permeability in marcomesh for oil or water phase.
After the initial definitions, we were able to develop an equation that represents the relative permeability in the macro mesh for the oil and water phases, according to the optimization of Eq. (20):
...,
...,
...,
...,
...
Observing, that:
1. У хех P(x) pOr |x) with respect to y, returns the probabilities associated with y, ie. 40);
2. Xxex P(X) р(ук|х) with respect to x, returns medium value, i.e., =;
So:
...
As the algorithm treats the probability of information regardless of the realizations, then in estimating them, we have:
...
So:
Foi oil phase:
...
For water phase:
...
Sidebar
References
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